 So thank you, Jan. It's great honor for me to speak at this wonderful conference. It's, of course, the perfect place for me to express my depth and gratitude to Maxime's ideas and influence. And when Maxime invented Motivique Integration almost 20 years ago, I'm not sure he really could anticipate how many developments there would be and the wide range of applications. So initially I plan to present a general overview of many of these developments, but finally I've chosen to focus on more specific recent results that combines two different topics. Maxime has contributed, so as a side note, in fact I have the privilege to share Maxime's birthday date. Of course, I'm not born the same year, so I know perfectly well which day he was born, and so I know when I will have to wish him a happy birthday. Okay, so this is joint work with Antoine Chambelois. It's available as a preprint and archive. And so the topic of the talk, if some is an application of Motivique Integration to curve counting, so this seems appropriate to talk, you know, of Maxime because he contributed both topics. However, the approach I will present is, I think, kind of new. And I must say that I think it is a method I will present that is maybe more interesting than the final result we have. I mean, I'm not so sure the result I will present is really interesting for himself, but maybe the method is. Okay, so we will make fundamental use of a result by Ruchowski on cashdown, which is so-called Motivique Poisson formula. So a substantial part of the talk will consist in making sense of this formula, what it means, and then I will explain how it can be applied. So now I just want to mention what Ruchowski on cashdown did with that, even if it has nothing to do with anything of this talk. So if you don't understand what's written on this slide, I mean, it's not a problem at all. And so they consider this following situation, so they start with a field of characteristic zero and they consider two division algebras of the field of rational functions on F, which are associated with two distinct elements of the same cyclic Galois group of prime order of the residue of F. And then you can compare data of Rd1 and data of Rd2. And so there is a notion of matching of integral conjugacy classes and of matching of local Motivique test functions, whatever this means. And what they wanted to prove is that if you start from a local test function that match, then they free transform match. This is what they want to prove. And what do they want to prove that? It's because a long time ago, I mean, there was a result by Deline, cashdown and Villeras over the PADICs. And they prove something similar. So it's an equality or a relation between PADIC integrals or local integrals. But it can only be proved at the time and still now by a global argument. So they needed use of the trace formula to prove this equality between local integrals. Trace formula? Yeah, yeah. And so, OK. There is a general philosophy that whenever you have an identity between PADIC integrals, there should be a similar identity between the Motivique ones. But this could be a possible counter example because the proof of the identity being not purely local and Motivique integration being apparently local, you could expect that this would provide a counter example. But in fact, the result of Rousseff-Key cashdown says that it does not provide such a counter example. OK. Now, the results I will present are a function field analog of some result in arithmetic. So I will start by recalling these results. So these are about Manin's conjecture on estimating the number of points of bounded height. So I think you all know what is a classical height. So you have a rational number and x ratio of a and b. Maybe b is non-zero. And so the height is just the maximum of the absolute values of both numbers. And so it's a measure of arithmetic complexity. So more generally, if you have a rational point in the projective space and you can consider the height by taking the point in homogeneous coordinate and with co-prime coefficients and you take the maximum of the absolute values. The third basic result is due to North scores. I mean, it says that if you fix b, then you have only finitely many points in the projective space with 8. This is good. That's all it is. This is completely empty. This is completely empty. Thank you. I'm sorry for that. I'm sorry. OK. What's less trivial is this, is that there is an estimate. Yeah. OK. Thank you. OK. And so as a... It's more or less obvious, yes. No, it's equally obvious. It's equally obvious. It's obvious, but it appears in the papers by these people. I'm sorry for that. No, maybe not the first one. No, but the trouble is you can talk to these people and put them in a rhythm by trying to understand what they're doing before. It's going to be different in another direction. But it won't. You have to figure out the issues, but it's going to be a problem. OK. OK. Most... I think everyone in this room is able to prove this theorem. The counts he proved. Yeah. OK. OK. And now I think maybe Don will be more happy. So we introduce a generating function. OK. And so it's called the Hay-Zeta function. And it encapsulates the asymptotic behavior of the number of these numbers. OK. We want to... So there's a general question by Manin. So, OK, it's a framework of number fields and function fields over finite fields. And you can replace pn by any projectivity. OK. Consider polarities. So, some... projectivity x. And the question is about understanding the asymptotic behavior of this counting function. So it should be of this form. And, OK, so Manin question is about understanding the exponent s, the exponent of b, and the t. I don't know what this looks like across, but it should be a t of the logarithm. Of course, this n may be also the c. So, of course, this is... when you translate into this generating series, this can be reinterpreted in terms of first pole and its order and so on by well-known Tauverian results. So you can ask about the abscissa of convergence of this function of s and about its possible monomorphic continuation and this pole. From the general answer, there's no spoilers for an editor solution to the human problem. Now, in pathological example, there is no advantage, right? Yes or no. I mean, I will add the hypothesis and... So what Manin proposes is to... in a quite special setting. So taking as ample bundle the opposite of the canonical divisor, so you're assuming, so it's the final framework, he gets a conjecture for what could be s and t. So s should be 1 and t should be the rank of the Nero-Severi group minus 1. But it's 45, right? Yes. But 45. And Perry finds this by being able to give a conjecture for the constant c as a kind of Tamagawa measure. So here one should be cautious to avoid trivial conterexamples. So okay, so as I said when one assumes x is Fano quite often, when it's better to have the rational points of x as a risky dance, otherwise the geometry of these rational points is governed by some sub-variative. So it has nothing to do with x. And sometimes it's also necessary to replace the ground field by a finite extension and also to restrict the counting problem because it could have strict sub-variatives with too many points and you should exclude these to recover some significant. So my name is a question, it's really a question, it's not the conjecture. So it's known in many cases, in some cases. So for hypersurface of small degrees, then it belongs really to analytical number theory. It uses the circle method. And one of the first cases known is that of flag varieties. So this uses a result by long glance on Eisenstein series and this is due to Frankel, Manin and Schinkel. Then there is a whole series of work dealing with equivalent compactification on some algebraic groups. So we started with Tori, so the compactification of Tori varieties. I find spaces, so this is Schambler-Lohr and Schinkel and generalized to simply connected semi-sample group by all these people and so on. And still another large direction is that of delpase surfaces and the list is continually growing. And there are also counter-examples, so it's not always true. So first counter-example is due to Batir-F Schinkel. It takes the total spaces of your universal family of diagonal cubic surfaces. This provides a counter-example. And you also have the Hilbert space of two points on P2 or P1 cross P1. But as of today, it seems that the power of S predicted by Manin always stands. I mean there is not yet a counter-example to that. The counter-examples are about the power of the logarithm and the constants. Okay, Bat. Now what is the geometric version of this? So this is a very classical dictionary. So we start with a projective curve, non-singular of a small k. And we consider its function field capital F. And so the geometry situation will be that of geometry of the curve C. And so rational point will correspond to a morphism from C to the projective space. Or if you consider Batir's relative projective space, it will be a section of this mapping. And the height or Batir's logarithm version, so small h with position to capital H, can be interpreted geometrically as the degree. So here you have the standard line model on Pn and you take its pullback on the curve and you consider its degrees. It is a number which is a logarithmic height. Okay, so more generally we will consider morphism from a variety to C and try to count sections of degrees smaller than something. This is the idea. Okay. Then in this setting finiteness of points will become the fact that, translate as a fact that if you consider a section of a given degree, they will not be finite but they will be formed or maybe better be the rational points of some algebraic varieties of a small k. So that's the idea. And now a pair in fact was the first to suggest around 15 years ago that one should investigate the behavior of these varieties when D varies with D. So this is the idea of the, this is what I want to explain today. Okay. So we start from, again we see, okay. And we'll have, so curly x will be just a proper flat morphism. Okay. We take some line bundle and when we restrict to the field of function, to the generic point of the curve, we put Roman letters instead of curly letters. And so we will assume that the codera dimension of L is maximum. And in fact, we will not really count all sections but we will, for technical reasons, we will have to add a condition of this type. So we will have to consider sections that belong to some appropriate subset U. So to guarantee that the set we want to compute really are constructible over the small field. Okay. For large negative degrees. So then we will consider the geometric A data function, which is a generating function. Okay. And we will work in a framework when the number of negative terms is finite. So it's really a long series. Okay. And we will use this as a formal long series with coefficient of the got generic of varieties. So let me now explain what is this ring. You all know, but in fact, since we will use Fourier transform, we will need to enlarge these classical rings to some larger one which are allowed to perform Fourier transform. Why are you asking? Why is it called motivic? So maybe you should ask a question to Maxime because he is the one who said motivic integration, who invented the terminal. He did not only invent the concept, but invented the terminology motivic integration. But historically, I mean, I think there is a good reason to call them motivic because they, this ring came out, they occur in the correspondence between Gotendic and Serre. Yeah, of course. And with the question of comparing them to actual motives, and if I remember correctly, Gotendic explained that these guys are being too complicated to understand. It's maybe better to consider motives which are more linear. This is explicitly in the correspondence between Gotendic and Serre. Okay? But you'll write that we won't see any actual motive in the strict sense. Okay. So the Gotendic ring of varieties of a field K is just defined by considering the free abelian group on isomorphism classes of algebraic varieties of K. And you add this physical relation. So whenever Y is closed in X, you ask that the class of X is the sum of the classes of Y and Y is complement. And you get also ring structure by just taking the Cartesian product of K. So it's a huge ring, in fact. It's complicated. It's known that there are zero devices. And usually when one does not work in this ring, one has to consider some localization of this ring. Okay? We will see the one we need later on. So there is a unit element, just a class of the point, and a very useful left-right element which is the class L of the affine line. And a trivial relation can express the class of the projective space in terms of this. Okay. So that's for the Gotendic ring of varieties. And so just as a warm-up, if we consider the pn cross c, and this genetic series, we look at sections of degree d, then in fact this was anticipated by Misha Kapanov in a paper which I think is unfortunately not yet published. A long time ago. In fact, he provides almost everything that was needed for pair to show that this series is rational and that to determine its largest pool. And even the residue, so which can be interpreted as, you take, so this limit, you multiply by the denominator. And it converges in some completion, which is a completion that Maxime introduced when he introduced the motivic integration. It's some completion of this localized group. So he gets the pool and the residue. And as a corollary, you can get estimates for the dimension, asymptotic dimension of these spaces md. So they grow very much like n plus 1d. So they have a finite limit. The difference is a finite limit. And if you consider the number of irreducible components of md of maximal dimension, you also get an estimate. So maybe just a few words to explain how you get this from the two previous statements. So you have many additive invariants of algebraic varieties. So one of them is so-called Hodg-Jolene polynomial, which is built up with Hodg numbers. And of course, you can recover the dimension of an algebraic variety from the Hodg-Jolene polynomial. And also the number of irreducible components of maximal dimension. So this can be read off from the Hodg-Jolene polynomial. And so by using the Hodg-Jolene polynomial, from knowing the order of the pool, the pool and its order, you are able to get these asymptotic results of dimension on number of components. So this is basic idea. Now, what we will do is the following. We will consider any irreducible smooth projective curve or an algebraically closed field of character zero. So it could be c, complex numbers. And we fix a non-empty open subset of c. And we take proper smooth variety over c. There's a non-constant morphic to c. Some line-bound along this curly x. And we fix also the risky open subset. And as before, we denote with romance the corresponding object under the algebraic fiber. And we make very strong assumptions. So our result is very, we deal with a very specific geometric situation which has some interest, but I don't claim it's very extremely interesting by itself. For instance, as I will explain at the end of the talk, it would be much more interesting to consider toric varieties instead of this. So we will consider a curiant compactifications of vector spaces. Okay? And so we assume this open u is just a vector space of rank n viewed as an additive group. So x should be a smooth and compactification of this. And we assume that the complement has strict normal crossings. And we take as L the logarithm version of minus canonical divisor. Okay, which is something very natural in this context. Okay? And now we will consider modular spaces of sections which are of degree d and that send this c0 which was a given open subset of c into u. Okay? This means that we don't consider in fact really a geometric analog of Manin's question. We don't deal with rational points but more with integral points. Okay? Because this is an integrality assumption, this condition. Okay? Now just by some general results on equivalent compactifications of the fine spaces one can prove that the space md exists at least as a constructible set. So it has a well-defined class in the quotient degree. And that for very negative d the md is empty. So we are in the right framework to go on. So we consider the generating functions. But so now what localization we will work in is so we not only invert L but we invert all classes La minus 1 when A is strictly positive. Or you could say, okay. And so we have such a Laurent series. So of course the very small ring is just these Laurent polynomials. And we will consider intermediate rings which essentially says that the series is the largest pole. So we have a variable t but it should be viewed as L to the minus s. Okay? And so we are just saying that so this m quality just says that the largest pole is at real part of s1. And this one just says that you can slightly continue to the left. Okay? So it's formally so you just locally you just consider the inverses of these polynomials 1 minus La Tb. But so only for b larger or equal to a. So as I said this corresponds to this condition if I'm not wrong. And the smaller one is you impose the stronger condition that b is strictly larger than a. Okay? And so this allows to evaluate such series here at L minus 1 just by beginning. Okay? You can evaluate them. This is obvious. And so our main result is that under the assumption I have given. So the series at t is a rational series but it belongs to m quality. So the largest pole is at t equals L to the power minus 1. And so here we just multiply by something related to the largest pole. And the p here is in empty dagger. So I can evaluate it at L minus 1 and I have really an actual residue. So I get something which is effective and on zero. So to speak strictly positive. In fact there is one can precisely compute this t but I won't in this lecture. And now if we look at the geometric consequences by one can get from this statement by using the orderly polynomial. So we get results on the asymptotic dimension of the behavior of the dimensions of the MDs. And also about their number of irreducible components. So there is this a here which is a bit, okay, a bit an artifact. But okay one can see one cannot really avoid it. So for any integer smaller than a, either the dimension of MD is a small o of d. Either you have something similar behavior as in the case of projective spaces. So minus d if you subtract d you get a finite limit. And also the ratio of the logarithm of K of MD and log d converges to some integer in this interval, okay. And the second one occurs at least one, at least one. So you get at least one p in this interval. For instance if a is one which is the best possible situation. You have only one p and you get this statement, okay. So these are the geometric consequences. And now this is based on the Poisson formula. So let me explain what the Poisson formula is. So the classical Poisson formula you all know what it is about. So you take a smooth function which is sufficiently decreasing at infinity. Okay this is, the heart is missing. And if you take the sum of the values of f on the integral point. It's equal to the sum of the values of, it's free transform. And more generally if you have a locally compact WN group. And it's point here again, dual. Take any discrete compact group. Then if you consider matching R measures on G and its dual. Then you get a similar statement. So here the sum is of the point in the discrete compact group. And the sum on the dual side is just, you model by gamma and you consider the dual which is same as considering the orthogonal, okay. And so for instance this works when the field small k is finite. Because then you have all the local compactness you can want. And the field f of rational function is discrete compact. It's the other group. So this works out directly. But of course when small k is capital C, this does not work up here. And so this is what I know I want to explain the Rousseff-Hicke-Jean formula. So it's of our curve. Take for simplicity in algebraic closed field k. G is a genus. And so it goes as follows. So I don't claim it, for the moment this has no meaning. But this is what I want to explain. So you take the sum of the value of a function at all points in fn. And a similar sum for the Fourier transform. And the equal modulo factor l to the power 1 minus gn. G is the genus of the curve. And the hypothesis is that, okay. But of course this is not yet defined. I will, it takes some time to define. So the space s is a space of so-called Motivix-Rath-Brouya function on the Adelic space. Here you get the Fourier transform. So this space should allow for some Fourier transform. And the sum is also, even if it looks very finite, it should be an element of some gotonic ring. Okay? So now let me explain the Rousseff-Hicke-Jean. So as Fourier transform is playing a role here, I should explain what, about how to deal with exponential. So if, so now it's a variation of the gotonic ring of varieties. So instead of just considering classes of varieties, you further consider a function f, a given function f on x. Okay? Any morphism f from x to the affine line. And so you get the similar, you consider similarly the c-cell relations, but you have one additional formula. I should explain. So take x is the point. And then you get that a1 with the function identity should be 0. And this should be interpreted by expressing that if psi is a non-trivial additive character of a finite field, then it's well known that if you add the values of psi on the affine line, you get 0. Okay? And so this is really the meaning of this additional relation. So what I found, so we introduced these rings with clockers and they were also considered independently by Rousseff-Hicke-Jean. And I want to say I was really amazed by the fact that this simple relation is enough to develop a theory of integration with exponential. Okay? This was not obvious for me at all at the beginning, but just this very simple natural relation is enough to do whatever you want. Okay? So xf should be sort of something like the function exponential of f on x. So this explains the ring structure. So if you have f on x and g on y on the product, you get exponential of f plus g, which explains this. And now it's still a ring. You have a unit element. And you have, in fact, an embedding of the classical one. So we can localize and you have an embedding of the classical one to the exponential one by just taking the function 0 whose exponential is 1. Okay? So that's the idea. Now we need to do everything relative. So instead of working over k, we work over some k-scheme, s. And so everything works out relatively. We have a functoriality. And so if you have a morphine between two bases, s and t, you can go from a function on t to a function on s just by pullback. So this amounts geometrically to Cartesian product. Or you can just integrate. So integrate, you have something over s, and you view it as being over t. So this looks very trivial, but this is what it amounts to, to integrating. Okay? So for instance, if you view x over itself, of course, there are much less isomorphisms that if you view x over spec k. And okay, so this operation is, even if it looks completely trivial, it means something. And as I said, so an element of such a ring, it can be thought of as a motivic function with exponentials. Okay. So I should speed up. Maybe not as much as Don did, but... So not take some work locally. So k is the radio field, and we consider the field of Laurent series and the ring of formal series. And I want to deal with Schwarz-Bruyer function. So Schwarz-Bruyer functions of the PADs are just function with compact support, and which are locally constant. So the picture here is the following. You have a large ball, and you have smaller balls. So here m and n are relative to the valuation radius of the balls. And so you want the function to be zero outside and to be constant on each ball. Of course, this amounts to having a function just in the set of small balls inside the big ball. And so, sorry, this... So, and this set of balls is just, can be just expressed as this quotient, which is, of course, just a set of rational points of the affine space. It's just saying how much digits you need to have to determine a small ball within a big ball. Okay. And so the good news is that this guy is an affine variety of finite type. And so the Schwarz-Bruyer function of level mn will just be our functions on this space. What else? Okay. And now the game is... So we can evaluate the integrals. So there is some normalization procedure just by taking... So this is just going to the point, pi. And now I don't have enough time, but okay, you can vary n on m. And if you have this picture in mind, everything what you can do is transparent. And so you can... You have various rows, and you can just define the Schwarz-Bruyer function as double direct limits of these spaces of functions of level mn. Okay. And you have a way to define integral which does not depend on the way you represent things. Okay. This is... Okay. It is what it should be. Okay. And now you can consider Fourier transforms because you have exponentials. Okay? A way to do that is to fix a non-zero global differential and to take residues. This is a nice way to get a billionaire form. And so you get in this framework Fourier transform, and you get a Fourier inversion. Okay? Now we should globalize that. So we... This was purely local, up to now. So we work over this curve c, f is function field, and for any rational point on the curve, we get a local field, localization. And we did the construction for given fv, but since they all are the same, we could consider also a finite product of this fv. Okay? So for given finite set, we can consider Schwarz-Bouillard function, not on one fv, but on the product of the fv. Okay? And you can do the same Fourier inversion and so on. And we can enlarge this, and we get an embedding just by, so to speak, adding to the new place the characteristic function of the ring of integrals, of the valuation line. And now we get a ring of global Schwarz-Bouillard function which are just defined by... which are just, in fact, non-trivial at the finite number of places. So this is enough for what we want to do. This is not conceptually satisfying because as you... I mean, for many applications, we need Schwarz-Bouillard functions that are non-trivial at all places. Okay? But since it's not obvious to consider infinite products algebraically... That's impossible. It's possible to make something... It depends on the whole also... Yeah, yeah, yeah, yeah, yeah. Things of that are considered by Oshofi Kajan but the most generality is still to... it's something open, I think. But, okay. So this is a very baby case, but which is enough for what we want. We can do Fourier transform. We have Fourier inversion again. And now to get a full Poisson formula, we still have to understand how to sum over the rational points. Okay? But... So the key point is that f is bad as an object of the ratio field, but it's an inductive limit of these spaces. So these are standard spaces, Riemann-Roch spaces. So for any, say, effective divisor d, you take Ld. And f is an inductive limit of these Ld, and these Ld are, of course, finite-dimensional K-vector spaces. Now given a global function phi, it may be represented by something like that. So you have some n and m depending on the place. And now if you take... So you have your divisor d. So for any finite set, you can evaluate a function in Ld at these places. And you can truncate, and you get morphine from Ld to these spaces parameterizing both. Okay? And so it's easy to check that when d is large enough, things stabilizes and stabilizes. So you can just say the sum of my function phi here on f. I just, for d large enough, take the pullback of... It's a function here. I take its pullback here on here, and just take its push forward to the point, and this will define the sum. Okay? And now I can state the Ruch-Refri-Kachdan Poisson formula. Can we say even the Ruch space can be just in the system? Yeah, yeah. It's a kind of vector space. Yeah, just the space of a rational function whose divisor is larger than minus d. So now you can make sense of this formula, which is what they prove. In fact, the formula is not trivial, but it's only relies on Riemann-Roch and seriality. So it's not trivial, but it's elementary. Okay. So in the last part of the talk, now I sketch the proof. Okay? So in the arithmetical proof, in the arithmetical case, okay, or as a geometric version of a result by Chamberlois and Schinkel in the arithmetical case. And so the classical things they do, I mean, to count parts of bounded 8, they first extend the 8 to a function on the Adelik space. This is the first step. And so in their case, I mean, Gf is compact discrete in the Adelik space, so they will use Poisson formula to get a new expression for the zeta function of the 8. Chamberlois in the form. Okay? And the key is, so now the parameters here are the characters. And so a special role is played by the trivial character, which is y equals 0, which is the main contribution. So, and this is the key. So you have a finite sum, but one point is given the main contribution, and the other points are given smaller contribution. So this is, here, one is really doing some, it's really free analysis, I mean. The use of Poisson formula allows just to extract the main term in this series. Okay? And so the main term is here, and one can compute everything almost explicitly. It's always down to classical Iguza zeta function. And in particular, one can get meromorphic continuation and the main pole. And then I want us to consider the remaining wise. So in fact, everything is 0 outside some lattice. You get meromorphic continuation with less poles, and in fact, you can sum over the whole lattice except 0 to get similar result, and this is enough to conclude. In fact, we can do exactly the same thing in the Motivy framework. So it's easy to say. In fact, in practice, we could do it, and what is strange is it really requires some work to perform, even if at each step it looks rather easy. And so here what can we do? So we localize the height as a sum of local intersection numbers. Our assumption of being integral is very important to only have to deal with the finite number of places. And this, we have to make this to assume that our Poisson formula only deals with situations which are non-trivial only at the finite number of places. And now we express our zeta function, so it's a generating series, and we make the Fourier transform of this. And so the key is that, of course, only a finite number of places play your role. And we have to understand these Fourier transforms, which now are local. And very much as in the arithmetical case, for the trivial character, you just get something like a motif with a zeta function, so you can understand it very classically. And for more general why, you get some oscillating integrals, but not too hard to deal with. For instance, you get something analog to this kind of integral, and so you have to understand that this kind of integrals are smaller poles than if you have no exponential under some assumption, and this is not hard in the period x, and it works also in the motivic case, for instance. You have to know things like, so if you integrate the exponential of 1 over x over an annulus of large order, then you get 0, and this is very much the same as the global homomorphic of this integral. So what's funny is that Chambelois and Schinkels, they really used some analytical arguments, so you have to bound certain things in some strips to use majorizations, complex analysis, and so on. And all these steps can be adapted via some maneuver to perform it in the motivic case. Okay, so as I said, the main term will be correspond to the trivial character, and the summation, in fact, can be restricted instead of a lattice to a finite dimensional vector space, and when the character varies in the lattice, you get rationality of individuals and estimates of the poles which can be preserved on some at least constructible partition of V, which is very much enough for motivic integration, and this is the way we prove the statement. So I will finish with a few concluding remarks, so of course it would be very relevant to be able to extend the framework to consider to deal with actual and finite products, and in particular to get motivic tamagran numbers. So there are some hints about this already, but I think one could understand these things better. Also to change the group, so for instance, to deal with tori, so this would understand motivically, it would mean understanding multiplicative characters. So there is a big difference between multiplicative characters and additive characters. So of a finite field, there is only one non-trivial additive character, so to speak, which allows you to identify the additive characters with the field itself. But for a multiplicative character, it's very different, I mean. They're not all the same. And so it's not at all clear what would be a motivic version of the character and what would be a genuine melin transform in this setting. And this issue is really what makes it too premature to consider a toric case, and okay, we can even try more general groups like Chinker and some of his collaborators did. Okay, this is done. Questions? Your question is, how is it related to critical values? Yeah, the answer is yes, yes. So you can put the story, you can relate them with the motivic, the fiber and so on. In the same way as in the classical case, the asymptotic behavior of exponential integrals are related to the local invariance of the singularity. So you can play the same game. Okay, and also, what I want to talk about is the multiplicative character that's considering these things of jam, this multiplicative convolution. Yeah, yeah, yeah. That is kind of a pretty formal object in a sense, yeah. But in principle, I had several years ago as the integrable system. Yes? It's kind of a form of pure materiality in this geometric setting, kind of strange commutative rings. Okay, I would be delighted to discuss with you about this. In fact, if you have a stationary phase, what kind of invariance do you get because you're working overseas? There's a big mistake. So the local Fourier transform, so if you look at it, the more you get swing from backwards. But you're not working overseas. Yeah, I mean, so I mean, okay, here I presented a very, quite a special framework because I deal only with Schwarz-Brüder functions which are rather trivial functions, but if you deal in a more general setting, you have exponential integrals over C of t. And if you consider a thing like that or maybe put to t, then you get all the information, all the complex information of all the singularities, yes. Yeah, so this is very similar to Le Mans stationary phase. So this is really into the picture, yes. Did I answer your question? It corresponds to the same case. Oh, yeah, that's the problem, yes.